Inserted: 22 mar 2023
Last Updated: 22 mar 2023
Survey paper. Draft for conference proceedings of the workshop "Anisotropic Isoperimetric Problems & Related Topics" - Rome, 5-9 September 2022.
We review recent results on the study of the isoperimetric problem on Riemannian manifolds with Ricci lower bounds. We focus on the validity of sharp second order differential inequalities on the isoperimetric profile of possibly noncompact Riemannian manifolds with Ricci lower bounds. We give a self-contained overview of the methods employed for the proof of such result, which exploit modern tools and ideas from nonsmooth geometry. The latter methods are needed for achieving the result even in the smooth setting. Next, we show applications of the differential inequalities on the isoperimetric profile, providing simplified proofs of: the sharp and rigid isoperimetric inequality on manifolds with nonnegative Ricci and Euclidean volume growth, existence of isoperimetric sets for large volumes on manifolds with nonnegative Ricci and Euclidean volume growth, the classical L\'evy-Gromov isoperimetric inequality. On the way, we discuss relations of these results and methods with the existing literature, pointing out several open problems.